Poincaré–Perron's method for the Dirichlet problem on stratified sets

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J. Math. Anal. Appl. 296 (2004) 504–520

www.elsevier.com/locate/jma

Poincaré–Perron’s method for the Dirichlet probleon stratified sets

Serge Nicaisea,∗, Oleg M. Penkinb

a Institut des Sciences et Techniques de Valenciennes, MACS, Université de Valencienneset du Hainaut Cambrésis, 59313 Valenciennes Cedex 9, France

b Voronezh State University, Universitetskaja pl. 1, 394000 Voronezh, Russia

Received 15 April 2003

Available online 6 July 2004

Submitted by M. Passare

Abstract

We extend standard Poincaré–Perron’s method to the Dirichlet problem on a class of multures. This method is based on the spherical mean theorem, the construction of fundamental sand on Harnack’s inequality on such domains, that we first establish. 2004 Elsevier Inc. All rights reserved.

Keywords:Perron’s method; Harnack’s inequality; Multistructures

1. Introduction

The study of partial differential equations on multistructures is an expanding fielda lot of applications in continuous mechanics, aerodynamics, biology, and others (sexample, [3]). As standard questions, let usquote: solvability, regularity of the solutionspectral theory, control problems, numerical approximations, see [2,3,5–8,11,17–and references cited therein.

Weak solvability of elliptic problems is now well understood and is usually baseso-called Poincaré’s inequalities [1,2,12–14,19,20,22]. Their strong solvability fordimensional multistructures can be deduced from the weak solvability due to the

This work was supported by Russian Foundation of Basic Researches: Grant 01-01-00417.* Corresponding author.

E-mail addresses:[email protected] (S. Nicaise), [email protected] (O.M. Penkin).

0022-247X/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2004.04.014

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 505

l-vior oftrongtrary-initialtifieds, and

atified

e

a

en

e

roblem

to this

of ho-

dimensional structure (for parabolic problemsee [4,6,18]). For higher-dimensional mutistructures, the situation seems to be more complicated due to the singular behathe solution [9,19,23]. Therefore in this paper we start the investigation of the ssolvability on a class of arbitrary-dimensional multistructures, namely on some arbidimensional stratified sets, using so-called Perron’s method (a modification of anidea of Poincaré). Even if the strong solvability is obtained for two-dimensional strasets, all the intermediate steps like the spherical mean theorem, maximum principleHarnack’s inequality are given in the general setting.

The easiest way to explain the concept of elliptic equation of second order on strsets may be given by the following example using classical Green’s formula. LetG be anopen region ofR2 with a smooth boundary∂G. According to classical Green’s formula wmay write∫

G

u∆v ds +∫∂G

u∂v

∂νdl =

∫G

v∆uds +∫∂G

v∂u

∂νdl, (1.1)

whereν is the unit interior normal to∂G andu,v are sufficiently smooth functions.If we introduce a measure on the unionΩ = G ∪ ∂G (it is the simplest example of

stratified set with two strataG and∂G) by the following expression:

µ(ω) = µ2(ω ∩ G) + µ1(ω ∩ ∂G), (1.2)

whereω ⊂ Ω , µ2 is the two-dimensional Lebesgue measure onG and µ1 is the one-dimensional measure on the boundary∂G. In term of that measure (1.1) may be rewrittin the form∫

G

u∆v dµ =∫G

v∆udµ, (1.3)

where∆ is defined as follows:

∆ =

∆ in G,∂∂ν

on∂G.

One of the most remarkable features of the operator∆ is that it may be represented in thform

∆u = ∇(p∇u), (1.4)

wherep ≡ 1 on G, p ≡ 0 on ∂G, the above expressionsp∇u and∇ F being correctlydescribed in the next section.

As we shall see below the above considerations are related to the solution of the p∆u = 0 in G,∂u∂ν

= 0 onΓN ⊂ ∂G,

u = g on∂G \ ΓN.

In the next sections such problem will be solved adapting standard Perron’s methodsetting.

The extension of our method to quasi-periodic stratified sets using techniquesmogenization from [28] will be done in forthcoming works.

506 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

tion 2,c-o theuctionn Sec-

someenition.r

at

n of

et

tse

.

The schedule of the paper is as follows. After recalling some basic notions in Secwe prove in Section 3 the spherical mean theorem forp-subharmonic functions. In Setion 4 some properties ofp-subharmonic functions are deduced. Section 5 is devoted tstatement of Harnack’s inequality on stratified sets which is deduced from the constrof fundamental solutions on some stratified balls. Finally all these results are used ition 6 to implement Perron’s method.

2. Exact formulation of the basic notions

This section is devoted to the exact description of the notions used later on and toauxiliary features. The general definition of stratified sets may be found in [24], but thstudy of differential equations on stratified sets needs some specifications of that defiNamely, by stratified set we mean a connected subsetΩ of R

n, consisting of a finite numbeof smooth submanifoldsσki (strata) ofRn which satisfy the two following conditions:

• the boundary of ak-dimensional stratumσki consists of (pairwise disjoint) strataσmj

with m < k.• If X ∈ σk−1i , then there exists a diffeomorphismΨ of some neighborhoodUX

of the pointX, which straightens the intersectionUX with the “star” S(σk−1i ) =σk−1i ∪ ⋃

σkjσk−1iσkj (the notationσmi σlj means thatσlj ⊂ σmi ), in other words

Ψ (S(σk−1i ∩ UX)) is a subset of the union of a finite number ofk-dimensional half-spaces, adjoining toeach other along some(k − 1)-dimensional edge.

The second condition is automatically satisfied, if all of the strata are polygons. Thcondition is fulfilled in the example shown in Fig. 1.

One setΩ admits a lot of stratifications (i.e., representations in the form of uniomanifolds). An exact definition is given in [14,22,27] and consists of a tripletΩ,S,Φ,whereS is a collection of subsets (strata) ofΩ andΦ describes the construction of the sΩ using the strata ofS. We do not go into the details and refer to [14,22,27].

As a subset ofRn the setΩ inherits the usual topology of that space. LetΩ0 be a subseof Ω , consisting of some strata ofΩ . We assumeΩ0 to be connected, open and denin Ω . Its complement∂Ω0 = Ω \ Ω0 is then the boundary ofΩ0 in terms of the abovementioned topology. The representation ofΩ in the formΩ0 ∪ ∂Ω0 is clearly not uniqueFor the example of Fig. 1, we can take the full geometrical boundary as the set∂Ω0, i.e.,

Fig. 1. Example of a stratified set.

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 507

e

aren

e

oticedt with

thelarncein

he

reen’s

not only the part drawn by the bold lines. From now on we assume that the set∂Ω0 isnonempty.

Let µkj be the usual Lebesgue measure onσkj . For a subsetω ⊂ Ω we can define its“stratified” measure by the formula

µ(ω) =∑σkj

µkj (ω ∩ σkj ). (2.1)

A setω is measurable with respect toµ if and only if all “traces”ω ∩ σkj are measurablwith respect toµkj in Lebesgue’s sense.

A vector field F will be called tangent toΩ0, if for eachσkj ⊂ Ω0 andX ∈ σkj thevector F(X) belongs to the tangent spaceTXσkj .

If the restrictions ofF to all strata are differentiable (we do not assume that theresome connections between different restrictions; for example,F may be not continuous othe full Ω0), then we define the divergence ofF at the pointX ∈ σk−1i as follows:

∇ F(X) = ∇k−1 F(X) +∑

σkj σk−1i

ν · F |kj (X). (2.2)

Here∇k−1 F(X) is the classical(k − 1)-dimensional divergence of the fieldF restrictedto σk−1i . The symbolF |kj (X) means the extension by continuity to the pointX, which is

supposed to exist. The set of fields with this property will be denoted byC1σ (Ω0). As usual

the vectorν at the pointX is the unit normal vector toσk−1i directed to the interior ofσkj .The above definition of the divergence operator is not artificial and is the full analogu

of the classical divergence, since it appears as a density of a flow generated byF withrespect to the stratified measure, more details may be found in [27]. It should be nthat the sum in (2.2) is empty for the strata whose interior points are not in contacother strata. For example, this is the case for the strata of maximal dimensiond = d(Ω).In such strata, classical and stratified divergence coincide.

Let u be a continuous function onΩ0. We say thatu ∈ C2σ (Ω0), if the gradient field∇u

belongs toC1σ (Ω0). Here we follow the physical tradition to denote the gradient and

divergence by the same symbol∇ since no confusion is possible: if it is applied to a scafunction it is the gradient operator, while if it is applied to a vector field it is the divergeoperator. Note that the gradient at the pointX ∈ σkj is interpreted as the usual gradientX of the restriction ofu to σkj .

Let us fix a functionp :Ω0 → R such thatp ≡ pkj on eachσkj , with pkj 0. Thenfor anyu ∈ C2

σ (Ω0) the expression∆pu ≡ ∇(p∇u) is well defined and corresponds to tdefinition of an elliptic operator of divergence type on the stratified set.

In the sequel we need the following assertion, which is a full analogue of second Gformula and whose proof may be found in [26].

Theorem 2.1.Letu,v ∈ C2σ (Ω0)∩C(Ω) be differentiable up the boundary∂Ω0 (again in

each stratum separately). Then∫Ω0

(u∆pv − v∆pu) dµ = −∫

∂Ω0

(u(p∇v)ν − v(p∇u)ν

)dµ. (2.3)

508 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

),

uation

f

m.

habout

The quantity(p∇u)ν at the pointX ∈ σk−1i ⊂ ∂Ω0 is defined as in the formula (2.2i.e.,

(p∇u)ν(X) =∑

σkj σk−1i , σkj ⊂∂Ω0

ν · p∇u|kj (X),

and is the analogue of the standard normal derivative multiplied byp.Let u be a solution of the equation∆pu = 0. Then we obtain from (2.3) (takingv ≡ 1)

the useful identity∫∂Ω0

(p∇u)ν dµ = 0. (2.4)

This formula has also an obvious classical analogue. Here below solutions of the eq∆pu = 0 will be calledp-harmonic.

3. Spherical mean theorem andp-subharmonic functions

Now we restrict ourselves to the case when all strataσkj (including those lying in∂Ω0)are flat, this means thatσkj lies in somek-dimensional plane of the spaceR

n. Moreover weassumep to be a positive constant within the strata of highest dimensiond = d(Ω) and tovanish in all other ones. Under these assumptions the operator∆p has some properties oclassical type. We call it a soft Laplacian in contrast to the casep > 0 everywhere, calledhard Laplacian. In this section we concentrateour attention to the spherical mean theore

Let X ∈ σkj ⊂ Ω0 andr0(X) be the distance betweenX and all σd−1i such thatX /∈σd−1i . Let Br(X) be the usual ball inRn of centerX and radiusr. Whenr < r0(X) wecall the set

Br (X) = Br(X) ∩ Ω0

a stratified ball. An example is shown in Fig. 2 and illustrates the general situation.Let us denote byMBr (X)(u) the weighted spherical mean

MBr (X)(u) =∫Br (X)

pudµ∫Br (X)

p dµ

of the functionu with respect toBr (X) and byMdBr (X)(u) the analogous mean witrespect todBr (X). The next assertion gives the exact analogue of Gauss’ theoremspherical means.

Fig. 2. Stratified ball.

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 509

uality

nte

-

e

zationase and

nott.

Theorem 3.1.Letu ∈ C2σ (Ω0) be ap-harmonic function. Then

MBr (X)(u) = MdBr (X)(u) = u(X).

Proof. We only prove the part of this assertion related to the spheredBr (X), the otherpart follows by integrating with respect to the radiusr.

Let us notice that the ballBr (X) has a natural stratification inherited fromΩ0. As aconsequence the identity (2.4) is applicable, if we takeBr (X) asΩ0 anddBr (X) as∂Ω0.Let us further notice that the restriction on the admissible radius of the sphere (ineqr < r0(X)) and the vanishing condition on the coefficientp on the strataσmj with m < d

lead top dµr+∆r = p(1 + ∆r/r)d dµr , wheredµr is the surface element (or the elemeof the stratified measure) of the sphere with radiusr. Using that circumstance and (2.4) wobtain the expression

∆MdBr (X)(u) = MdBr+∆r (X)(u) − MdBr (X)(u)

= ∆r∫dBr (X)

p dµr

∫dBr (X)

(p∇u)ν dµr + o(∆r) = o(∆r), (3.1)

when∆r is sufficiently small.The formula (3.1) leads toMdBr (X)(u) ≡ const when 0< r < r0(X) and the desired

assertion may be obtained by taking the limit asr → 0. In an analogous way one can prove that the inequality∆pu 0 implies u(X)

MBr (X)(u) as in the classical case. Instead of (2.4) we here use the inequality∫∂Ω0

(p∇u)ν dµ 0,

which is satisfied in the case∆pu 0. So, we can define the class ofp-subharmonic functions which consists of functionsu ∈ C(Ω) satisfying the inequalityu(X) MBr (X)(u)

for all X ∈ Ω0 and all sufficiently small ballsBr (X). The next section is devoted to thdescription of some properties of such functions.

4. Main properties of p-subharmonic functions

Here we present all properties of subharmonic functions that we need for the realiof Poincaré–Perron’s method. The main part of them are proved as in the classical care therefore omitted.

The spherical mean theorem leads immediately to the weak maximum principle.

Lemma 4.1.Letu be ap-subharmonic function. Then

maxX∈Ω

u(X) = maxX∈∂Ω0

u(X).

It should be noted that the strong maximum principle in its usual formulation isfulfilled for p-subharmonic functions. Nevertheless, we can give the following varian

510 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

l

lt

ppose

e

-

y

thee

hat

Lemma 4.2.Let u be ap-subharmonic function. Thenu has no points of local nontriviamaximum inΩ0.

Here we say that a pointX ∈ Ω0 is a point of local nontrivial maximum if it is a locamaximum and for each neighborhoodU of that point the functionu is not identical constanin U .

Proof. The proof of the above lemma is made by contradiction. Indeed let us suthat X ∈ Ω0 is a point of nontrivial local maximum ofu. Then in any ballBr (X) thereexists a pointY such thatu(Y ) < u(X). Therefore by the continuity ofu for all sufficientlyclose pointsZ of Y in Br (X) we haveu(Z) < u(X). This fact clearly implies that (for thfunctionu ≡ u(X))

u(X) = MBr (X)(u) > MBr (X)(u)

for sufficiently smallr. This is in contradiction with the definition ofp-subharmonic function.

Obviously the same assertion is valid forp-harmonic functions.

Lemma 4.3.Letu1, . . . , uk bep-subharmonic functions onΩ . Then the function

u(X) = maxu1(X), . . . , uk(X)

,

which we call an upper envelope of the familyui (i = 1, . . . , k) is alsop-subharmonic.

The next property may serve as an alternative definition ofp-subharmonic functions.

Lemma 4.4.Letu be ap-subharmonic function andv a p-harmonic function which satisfu v on the boundary∂Ω0. Then this inequality is satisfied inΩ .

Proof. The differenceu − v is ap-subharmonic function with nonpositive values onboundary. Taking into account the weak maximum principle the difference is nonpositiveverywhere inΩ .

Our last assertion is related to the so-calledp-harmonic cats. Letu be ap-harmonicfunction. Let us fixX ∈ Ω0 lying in some stratumσkj with k d − 1 andr < r0(X).Considerv the solution of the Dirichlet problem∆pv = 0 in the ballBr (X) with values ofthe functionu on dBr (X) as a Dirichlet condition (in the next section we shall show tthis problem is solvable). Now we define thep-harmonic cat

uX,r(Z) =

u(Z), Z /∈ Br (X),

v(Z), Z ∈ Br (X).

Lemma 4.5.Let u be ap-subharmonic function. Then anyp-harmonic catuX,r is alsop-subharmonic.

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 511

t

by

rss

bably

by the

y) aliesonular

einatesl

fown

Proof. It suffices to prove the inequality

uX,r(Z) MBρ (Z)(uX,r)

only at the pointsZ ∈ dBr (X), because forZ ∈ Br (X) the desired inequality follows fromthe spherical mean theorem and forZ ∈ Ω0 \ Br (X) it is a consequence of the fact thauis ap-subharmonic function.

For Z ∈ dBr (X) by the fact thatu is a p-subharmonic function and secondlyLemma 4.4 we have successively

uX,r(Z) = u(Z) MBρ(Z)(u) MBρ (Z)(uX,r ),

which proves the requested assertion.

5. Harnack’s inequality

5.1. Fundamental solutions

We are only able to give the fundamental solution of∆p for the spheresBr (X) withcenter lying in the strataσkj with k d − 1. But this is sufficient for the realization of ouPoincaré–Perron’s method since the coefficientp vanishes in the strata of dimension lethand .

The main obstacle to find the fundamental solution for an arbitrary sphere is prothe existence of singularities of the solution of the Dirichlet problem in the interior ofΩ0as detailed in [19]. As a consequence, the fundamental solution may not be foundmethod of reflections applied here.

A function G(Z,Y ) will be called a fundamental solution if∆pG = 0 with respect ofthe first variableZ whenZ = Y and additionally

φ(Z) =∫

Br (X)

G(Z,Y )∆pφ(Y ) dµ (5.1)

for any test functionφ ∈ C2σ (Ω0) with support inBr (X).

Let us denote byK(z, y) the classical Green function for the sphereBr(0) in Rd . This

function will allow us to build the fundamental solution which will be (simultaneouslGreen function inBr (X) with centerX ∈ σd−1i ; the case when the center of the spherein a d-dimensional stratum is trivial because inthis case it is the classical Green functi(up to a multiplicative factor). Moreover our formula below will restitute it as a particcase.

Let σdj1, . . . , σdjm be the strata adjoining toσd−1i . For our convenience writej1 =1, . . . , jm = m. We introduce some special coordinates onBr (X) (they are quite strangbecause that set is not a manifold). For this purpose we define at first coordx1, . . . , xd−1 on the stratumσd−1i and then for eachσdj σd−1i we define the additionacoordinatexd . We suppose that this additional coordinate vanishes onσd−1i and is posi-tive on each stratumσdj σd−1i . Besides we assume thatX is the center of that system ocoordinates (i.e., coordinates of the pointX are equal to zero). Such coordinates are shin Fig. 2 in the cased = 2.

512 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

ations

of

la

5.3)e

there

e

Let Y ∈ σdl ∩ Br (X), then we define the functionG(Z,Y ) as follows:

G(Z,Y ) = 2

P

pl+Pl

2plK(z, y) + pl−Pl

2plK(z, y), Z ∈ σdl,

K(z, y), Z ∈ σdj (j = l),(5.2)

wherez = (z1, . . . , zd), y = (y1, . . . , yd) are the coordinates of the pointsZ andY , re-spectively,z = (z1, . . . , zd−1,−zd), P = ∑m

j=1 pj andPl = P − pl .It is easy to see that the coefficients in front ofK(z, y) andK(z, y) (in the first line

of (5.2)) jump whenY passes from one stratum to another. This renders some calculmore difficult. Nevertheless, the functionG is continuous whenY = Z. Indeed, whenYapproaches toσd−1i the distances dist(z, y) and dist(z, y) become equal and the valuethe first line of (5.2) tends toK(z, y).

Theorem 5.1.G(Z,Y ) is a Green function of the operator∆p in the stratified ballBr (X).

Proof. ObviouslyG(Z,Y ) vanishes on the boundarydBr (X) with respect toY andZ,since bothK(z, y) andK(z, y) vanish on∂Br(0). The identity (5.1) is more difficult toestablish but follow a classical scheme based on Green’s formula. Indeed Green’s formuin Br (X) \ ωε , whereωε is a small spherical neighborhood with centerZ and radiusεyields ∫

Br (X)\ωε

G∆pφ dµ =∫

Br (X)\ωε

φ∆pGdµ +∫

∂ωε

(φ(p∇G)ν − G(p∇φ)ν

)dµ. (5.3)

Notice that the integral over the boundary of the ballBr (X) is absent, becauseφ has asupport in the interior of that ball. Now the first integral in the right-hand side of (also vanishes, becauseG is p-harmonic inBr (X) \ ωε . It is not so difficult to see that thintegral of the functionG · (p∇φ)ν over the boundary∂ωε tends to zero whenε → 0. Soour main problem is to investigate the integral∫

∂ωε

φ · (p∇G)ν dµ.

Assume thatZ lies in σdl . Taking ε > 0 sufficiently small we can also assume thatsphereωε is included inσdl and then looks like a usuald-dimensional sphere. Therefoinstead of the notation(p∇G)ν we may use the more traditional representationpl(∂G/∂ν)

of this quantity, whereν is the exterior normal toωε . Using the representation (5.2) of thfunctionG we can write∫

∂ωε

φ · (p∇G)ν dµ = 2pl

P

∫∂ωε

φ∂

∂ν

(pl + Pl

2pl

K(z, y) + pl − Pl

2pl

K(z, y)

)dµ,

what can be rewritten in the form∫φ · (p∇G)ν dµ =

∫φ

∂

∂νK(z, y) dµ + pl − Pl

P

∫φ

∂

∂νK(z, y) dµ.

∂ωε ∂ωε ∂ωε

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 513

t

the

ke

gration

centertifi-s

la (5.2)

hen theg

s

The second integral in the right-hand side tends to zero whenε → 0, becauseK(z, y)

has no singularities in the setωε . The first part tends toφ(Z), because it is well known tha∫∂ωε

φ∂K

∂νdµ → φ(Z)

whenε → 0. In conclusion the identity (5.1) may be obtained from (5.3) by takinglimit as ε → 0 and recalling the continuity ofG whenY = Z.

The additional caseZ ∈ σd−1i is similarly treated, but in the final part we must tainto account the fact thatωε intersects alld-dimensional strataσdj σd−1i . Since the co-efficients in the Green function are dependent of the strata we must perform an inteover∂ωε separately for each intersection∂ωε ∩ σdj .

It should be noticed that the expression (5.2) is also valid in the case when theX of the stratified ball lies in ad-dimensional stratum. In this case we use the arcial stratification into two hemispheres and the equatorial(d − 1)-dimensional plane a(d − 1)-dimensional stratum withp = 0 on this artificial(d − 1)-dimensional stratum. Inthis setting the expression (5.2) has the usual form since we easily see that formucoincides with the classical representation of the Green function.

One can also remark that the expression (5.2) is also appropriate in the case wcenter lies on a(d−1)-dimensional stratum with only oned-dimensional stratum adjoininto it, in this case the stratified ball looks like a usual hemisphere.

5.2. Poisson’s formula in stratified balls

If the support of the functionφ is not necessary in the ballBr (X), then considerationsimilar to those presented in the previous section lead to

φ(Z) =∫

Br (X)

G(Z,Y )∆pφ(Y ) dµ +∫

dBr (X)

φ(Y )∂G(Z,Y )

∂νdµ

instead of (5.1). Hereν is the exterior normal todBr (X). Takingφ ≡ 1 we obtain∫dBr (X)

∂G(Z,Y )

∂νdµ = 1. (5.4)

If additionallyφ is p-harmonic inΩ0, then

φ(Z) =∫

dBr (X)

φ(Y )∂G(Z,Y )

∂νdµ.

This circumstance justifies the name Poisson’s kernel for

P(Z,Y ) = ∂G(Z,Y ).

∂ν

514 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

tlently,

tion

whenel is

aly

In the sequel we need an explicit expression for the functionP . For this purpose isuffices to derive the expression (5.2) with respect to the exterior normal (or, equivawith respect to the radial variable). Recalling that

K(z, y) = Γ(|z − y|) − Γ

( |z|r

∣∣∣∣ r2

|z|2z − y

∣∣∣∣)

,

whereΓ (t) = 1

(2−d)ωdt2−d if d > 2,

Γ (t) = 12π

logt if d = 2,

ωd being the surface area of thed-dimensional unit sphere, we deduce after differentia

P(Z,Y ) = 2(r2 − |z|2)Pωd r

pl+Pl

2 |z − y|−d + pl−Pl

2 |z − y|−d, Z ∈ σdl,

pj |z − y|−d, Z ∈ σdj (j = l).(5.5)

Note again that the expression of Poisson’s kernel coincides with the classical onethe center of the sphere lies in ad-dimensional stratum. Note also that Poisson’s kerna nonnegative function (since|z − y| |z − y|).

Now we are ready to prove the basic assertion of this section.

Theorem 5.2.Let g :dBr (X) → R be a continuous function on∂Ω0. Then the functionudefined by

u(Z) =

g(Z), Z ∈ dBr (X),∫dBr (X)

P (Z,Y )g(Y ) dµ, Z ∈ Br (X),(5.6)

is a solution of the Dirichlet problem

∆pu = 0, (5.7)

u|dBr (X) = g, (5.8)

in the stratified ball with centerX lying in any strataσkj with k d − 1.

Proof. The fact thatu satisfies Eq. (5.7) follows from the fact thatG(Z,Y ) satisfies thesame equation with respectZ and from the possibility to differentiate under the integrsign. So, it remains to prove the continuity of the functionu at the points of the boundarof Br (X).

Let Z0 ∈ dBr (X) andZ ∈ Br (X) and consider the differenceu(Z) − u(Z0) = u(Z) −g(Z0). According to (5.4),u(Z) − g(Z0) may be rewritten as

u(Z) − g(Z0) =∫

dBr (X)

P (Z,Y )(g(Y ) − g(Z0)

)dµ.

Sinceg is continuous, for anyε > 0 there existsδ > 0 such that|g(Y ) − g(Z0)| < ε when|Y − Z0| < δ. Consequently we may write

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 515

lof

ly theion onar the

tymen-

g in

.

f

5)

∣∣u(Z) − g(Z0)∣∣

∫|Y−Z0|<δ

P (Z,Y )∣∣g(Y ) − g(Z0)

∣∣dµ

+∫

|Y−Z0|δ

P (Z,Y )∣∣g(Y ) − g(Z0)

∣∣dµ,

using the positiveness of Poisson’s kernel. The first integral is less thenε. The secondone tends to zero asZ tends toZ0, because of the factorr2 − |z|2 in Poisson’s kerneand because of the inequality|z − y| δ/2 (this inequality guarantees the absencesingularities) whenZ is sufficiently close toZ0. Therefore we have|u(Z) − g(Z0)| < 2ε

whenZ is sufficiently close toZ0. 5.3. Harnack’s inequality

Contrary to the classical case we cannot prove Harnack’s inequality using onspherical mean theorem for harmonic functions. The main obstacle is the restrictthe radius of the stratified balls (that can be very small even for the centers not neboundary). As a consequence we must prove the spherical variant of Harnack’s inequaliusing Poisson’s formula. Using this spherical variant and strong maximum principletioned above, we can prove general Harnack’s inequality for any compact set inΩ0. Butfor our purpose it is sufficient to prove it only for the stratified balls with centers lyinthe stratumσkj with k d − 1.

A spherical variant of the Harnack’s inequality is contained in the following lemma

Lemma 5.1.Let X ∈ σkj ⊂ Ω0 with k d − 1 and r < r0(X). Let u be a nonnegativep-harmonic function inΩ0. Then there exist two positive constantsC1,C2 independent ou (we give below the exact expression of these constants) such that for allρ < r we have

C1(r − ρ)rd−2

(r + ρ)d−1 u(X) u(Z) C2(r + ρ)rd−2

(r − ρ)d−1 u(X) (5.9)

for eachZ ∈ Bρ(X) such that the distance betweenZ andX is equal toρ.

Proof. Using Poisson’s formula we obtain

u(Z) =∫

dBr (X)

P (Z,Y )u(Y ) dµ.

The distance dist(Z,Y ) betweenZ andY may be estimated as follows:

r − ρ dist(Z,Y ) r + ρ,

and the same estimate is valid for dist(Z,Y ), because dist(Z,Y ) = ρ. It follows that|z−y| and|z −y| are betweenr −ρ andr +ρ. From that fact and from the formula (5.we obtain the following estimate of Poisson’s kernel:

2(r2 − ρ2)pj

d P(Z,Y ) 2(r2 − ρ2)pj

d.

Pωd r(r + ρ) Pωd r(r − ρ)

516 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

ve).

to

m

ed on

f a

Let X ∈ σd−1i and letσd1, . . . , σdk be all d-dimensional strata adjoining toσd−1i (ifX belongs to ad-dimensional stratum use the artificial stratification mentioned aboMultiplying the above inequality byu(Y ) and integrating overdBr (X) ∩ σdj we obtain

2(r2 − ρ2)

Pωd r(r + ρ)d

∫dBr (X)∩σdj

pju(Y ) dµ ∫

dBr (X)∩σdj

P (Z,Y )u(Y ) dµ

2(r2 − ρ2)

Pωd r(r − ρ)d

∫dBr (X)∩σdj

pj u(Y ) dµ.

Summing all these inequalities and taking into account Poisson’s formula (appliedthe sum of the middle integrals) we get to

2(r2 − ρ2)

Pωd r(r + ρ)d

∫dBr (X)

pu(Y ) dµ u(Z)dµ 2(r2 − ρ2)

Pωd r(r − ρ)d

∫dBr (X)

pu(Y ) dµ.

Summing these inequalities and taking into account Poisson’s formula applied to the suof middle integrals we obtain

2(r2 − ρ2)

pmaxPωd r(r + ρ)d

∫dBr (X)

pu(Y ) dµ u(Z)

2(r2 − ρ2)

pminPωd r(r − ρ)d

∫dBr (X)

pu(Y ) dµ.

The integrals on the left-hand side and on the right-hand side may be transformthe basis of the spherical mean theorem. As a result we have

2(r2 − ρ2)

Pωd r(r + ρ)du(X)

∫dBr (X)

p dµ u(Z)dµ 2(r2 − ρ2)

Pωd r(r − ρ)du(X)

∫dBr (X)

p dµ.

Remarking that∫dBr (X)

dµ = kωdrd−1

2

and making a rough estimate of the coefficientp (pmax = maxp1, . . . , pk, pmin =minp1, . . . , pk), we obtain

pmink(r − ρ)rd−2

P(r + ρ)d−1u(X) u(Z) pmaxk(r + ρ)rd−2

P(r − ρ)d−1u(X),

what should be proved.A standard consequence is the analogue ofHarnack’s theorem about convergence o

sequence of harmonic functions. Namely, the following theorem holds.

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 517

n

e

larif allge: we

ecase

Lemma 5.2.Let un be a nondecreasing sequence of harmonic functions inΩ0 which isconvergent at the pointX ∈ σkj ⊂ Ω0 with k d − 1. Then it converges uniformly to aharmonic function in some ballBr (X).

6. Realization of Poincaré–Perron’s method

Let us denote bySg the set of all subharmonic functionsu in Ω , which satisfy theinequalityu g on the boundary∂Ω0. Perron’s solution of the problem

∆pu = 0, (6.1)

u|∂Ω0 = g (6.2)

is a functionu which is an upper envelope of the setSg , i.e.,

u(X) = supv∈Sg

v(X). (6.3)

As usual a classical solution of the problem (6.1)–(6.2) is a functionu ∈ C2σ (Ω0) ∩

C(Ω) which satisfies these identities in the usual sense.For standard domains the proof that Perron’s solutionu satisfies (6.1) and (6.2) in th

usual (classical) sense consists of two independent stages. First we prove thatu satisfies thepartial differential equation and secondly that it satisfies the boundary conditions at regupoints (see, for instance, [16] or [15]), so that Perron’s solution is a classical solutionboundary points are regular. In the case of stratified sets there exists one more stahave to prove thatu is continuous in small-dimensional strata. This additional step will bdone on the basis of the investigation of the singularities of the solution but only in thed = 2.

It should be noticed that even ifd = 2, u may not be continuous inΩ0. For example, thesituation shown in Fig. 3 gives an example whereu may be discontinuous atσ01. Indeed ifg = 0 at the left part of the boundary∂Ω0 andg = 1 at the right part, then the solutionu isclearly equal to 0 at the left part ofΩ0 and equal to 1 at the right part.

To avoid this situation we need one geometrical restriction on the structure ofΩ and itspartition intoΩ0 and∂Ω0.

Definition 6.1.Ω0 is called firmly connected if∂Ω0 is contained in the closure of(d − 1)-dimensional strata lying in∂Ω0 and for eachσkj with k < d −1 the setΩ \σkj is connectednearσkj .

Fig. 3. Unsolvable case.

518 S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520

ce

ethe

es

[15]).dary is

ion

ta, in

e

art

eared

Let us start with the arbitrary dimensiond .

Theorem 6.1.Let Ω0, ∂Ω0 be a firmly connected pair andg is continuous. Thenu isp-harmonic and satisfies(6.2).

Proof. Let X be an arbitrary point ofΩ0, lying in some stratumσkj with k d − 1. Letus show that at this pointu satisfies (6.1).

Fix some stratified ballBr (X) and a countable setX1, . . . ,Xk, . . . , which is densein Br (X). From the definition of the functionu it follows that there exists a sequenui

k of subharmonic functions, which is convergent tou(Xi). Using the family of se-quencesui

k we can construct a nondecreasing sequence which is convergent tou on theset of all pointsXk . For this purpose it suffices to takevk = maxuk

1, . . . , ukk. According to

Lemma 4.3 that sequence consists of subharmonic functions fromSg . Taking the sequencof harmonic functionswk = (vk)X,r according to Lemma 4.5 it keeps all properties ofinitial sequencevk including convergence tou in the pointsXk .

According to Lemma 5.2 the limit function of the sequencewk will be harmonic oneach ballBρ(X) with ρ < r, and this function coincides withu in those pointsXk being inthat ball. Using the density of the setXk it is easy to see that the limit function coincidwith u everywhere inBρ(X).

To prove thatu satisfies (6.2) we argue as in the classical case (see, for example,Namely the existence of local barriers is automatically guaranteed since the bounassumed to be piecewise flat.

Now we restrict ourselves to the cased = 2.

Theorem 6.2.Let Ω0, ∂Ω0 be a firmly connected pair such that its maximal dimensd = 2 and suppose thatg is continuous. Thenu is a classical solution of(6.1)and (6.2).

Proof. According to Theorem 6.1, it remains to prove the continuity at the 0-d straother words at any corner pointS of a 2-d stratum. Fix such a pointS and consider a(radial) cut-off functionη equal to 1 in a neighborhood ofS and equal to zero near thexternal boundary and near the other 0-d strata. Thenv = ηu is continuous except atS andis solution of

p∆v = ηp∆u + 2p∇η · ∇u + p∆ηu = 2p∇η · ∇u + p∆ηu in Ω0,

and satisfies the Dirichlet boundary condition on the boundary ofΩ0. As 2p∇η · ∇u +p∆ηu belongs toH−1(Ω0), there exists a uniquev0 ∈ H 1

0 (Ω0) (weak) solution of∫Ω0

p∇v0 · ∇w = −∫Ω0

(2p∇η · ∇u + p∆ηu)w, ∀w ∈ H 10 (Ω0).

By Theorem 2.27 of [19] (see also [21,23]),v0 admits a decomposition into a regular pvR and a singular partvS . The regular part is piecewiseH 2 and is “continuous” throughthe 1-d strata so thatvR ∈ C(Ω0). On the other hand the singular part is a finite lincombination of functions of the formrλφ(θ), where(r, θ) are polar coordinates center

S. Nicaise, O.M. Penkin / J. Math. Anal. Appl. 296 (2004) 504–520 519

n is

)he

th-

cture

ns 72

93.artial

s

E,

lu-

00)

de

ger,

Multi-

r. A 291

g,

)

atS, λ is a positive real number andφ is a continuous function. ThereforeuS is clearly inC(Ω0). Altogether this means thatv0 belongs toC(Ω0).

Furthermorev − v0 is in C(Ω0), is equal to 0 on the boundary and by constructiop-harmonic. Therefore by the weak maximum principle

v = v0,

which proves thatv belongs toC(Ω0) as well. For d 3, the question whether the singularities of the weak solution of problem (6.1

and (6.2) are continuous is still an open problem, that is the reason of our restriction to tcased = 2. Indeed the weak solution has singularities along thek −d strata withk d −2[9,10,23] and their continuity is not yet clear.

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